A student claims that 1/8 is greater than 1/4 because 8 is greater than 4. What is wrong with this reasoning?
ANothing — larger denominators do mean larger fractions
BThe student confused the numerator and denominator positions
CThe denominator tells you how many equal pieces the whole is cut into, so more pieces means each piece is smaller — making 1/8 less than 1/4
DYou cannot compare fractions unless the numerators are the same
The denominator tells you the size of each piece, and the relationship is inverse: cutting a whole into 8 pieces gives smaller slices than cutting it into 4 pieces. So 1/8 is one small slice while 1/4 is one larger slice — 1/8 < 1/4. The student applied whole-number thinking (bigger number = bigger value) to a part that describes size, not count. This is the most common fraction comparison error.
Question 2 Multiple Choice
Which of the following strategies correctly compares 3/4 and 5/6 without converting to decimals?
ACompare numerators: 5 > 3, so 5/6 > 3/4
BCompare denominators: 6 > 4, so 3/4 > 5/6
CFind a common denominator of 12: rewrite as 9/12 and 10/12; since 10 > 9, then 5/6 > 3/4
DBoth fractions are close to 1, so they are equal
Finding a common denominator (12) rewrites both fractions with the same-sized pieces, making the comparison straightforward: 9 twelfths vs. 10 twelfths — 10/12 > 9/12, so 5/6 > 3/4. Option A fails because you cannot compare numerators when denominators differ. Option B applies inverted denominator logic incorrectly — larger denominator means smaller pieces, but you also need to account for how many pieces you have. Neither shortcut works here; the common denominator method does.
Question 3 True / False
When two fractions have the same numerator, the fraction with the smaller denominator is the greater fraction.
TTrue
FFalse
Answer: True
With equal numerators, you are comparing equal numbers of pieces, so piece size determines which is greater. Smaller denominator = larger pieces. For example, 2/3 vs. 2/5: both have 2 pieces, but thirds are larger than fifths, so 2/3 > 2/5. This is the 'same numerator' strategy — a reliable shortcut that depends on understanding what the denominator represents.
Question 4 True / False
The most reliable way to compare any two fractions is to look at which fraction has the larger denominator.
TTrue
FFalse
Answer: False
Larger denominators mean smaller pieces — so a larger denominator does NOT automatically mean a larger fraction. This strategy only works in a specific case (same numerator), and even then the logic is inverse: the larger denominator gives the *smaller* fraction. For fractions with different numerators and denominators, you need to find a common denominator, use benchmark fractions, or use another strategy that accounts for both parts of the fraction.
Question 5 Short Answer
Why does a larger denominator produce smaller pieces, and how does this affect which fraction is greater when the numerators are the same?
Think about your answer, then reveal below.
Model answer: The denominator tells you how many equal pieces the whole is divided into. More pieces means each piece is smaller — just as cutting a pizza into 8 slices gives smaller slices than cutting it into 4. When numerators are equal (same number of pieces), the fraction with smaller pieces is the smaller fraction. So 1/8 < 1/4 because eighths are smaller than fourths.
This inverse relationship — larger denominator, smaller piece — is the core insight of fraction comparison. It runs against whole-number intuition and is the source of the most persistent comparison errors. Understanding *why* the denominator is inverse (more cuts = smaller pieces) lets students reason through any same-numerator comparison correctly instead of guessing.